ARDL Models - Part II - Bounds Tests

[Note: For an important update of this post, relating to EViews 9, see my 2015 post, here.]

Well, I finally got it done! Some of these posts take more time to prepare than you might think.

The first part of this discussion was covered in a (sort of!) recent post, in which I gave a brief description of Autoregressive Distributed Lag (ARDL) models, together with some historical perspective. Now it's time for us to get down to business and see how these models have come to play a very important role recently in the modelling of non-stationary time-series data.

In particular, we'll see how they're used to implement the so-called "Bounds Tests", to see if long-run relationships are present when we have a group of time-series, some of which may be stationary, while others are not. A detailed worked example, using EViews, is included.

First, recall that the basic form of an ARDL regression model is:

           y_t = β₀ + β₁y_t-1 + .......+ β_ky_t-p + α₀x_t + α₁x_t-1 + α₂x_t-2 + ......... + α_qx_t-q + ε_t ,      (1)

where ε_t is a random "disturbance" term, which we'll assume is "well-behaved" in the usual sense. In particular, it will be serially independent.

We're going to modify this model somewhat for our purposes here. Specifically, we'll work with a mixture of differences and levels of the data. The reasons for this will become apparent as we go along.

Let's suppose that we have a set of time-series variables, and we want to model the relationship between them, taking into account any unit roots and/or cointegration associated with the data. First, note that there are three straightforward situations that we're going to put to one side, because they can be dealt with in standard ways:

1. We know that all of the series are I(0), and hence stationary. In this case, we can simply model the data in their levels, using OLS estimation, for example.
2. We know that all of the series are integrated of the same order (e.g., I(1)), but they are not cointegrated. In this case, we can just (appropriately) difference each series, and estimate a standard regression model using OLS.
3. We know that all of the series are integrated of the same order, and they are cointegrated. In this case, we can estimate two types of models: (i) An OLS regression model using the levels of the data. This will provide the long-run equilibrating relationship between the variables. (ii) An error-correction model (ECM), estimated by OLS. This model will represent the short-run dynamics of the relationship between the variables.

1. Now, let's return to the more complicated situation mentioned above. Some of the variables in question may be stationary, some may be I(1) or even fractionally integrated, and there is also the possibility of cointegration among some of the I(1) variables. In other words, things just aren't as "clear cut" as in the three situations noted above.

What do we do in such cases if we want to model the data appropriately and extract both long-run and short-run relationships? This is where the ARDL model enters the picture.

The ARDL / Bounds Testing methodology of Pesaran and Shin (1999) and Pesaran et al. (2001) has a number of features that many researchers feel give it some advantages over conventional cointegration testing. For instance:

• It can be used with a mixture of I(0) and I(1) data.
• It involves just a single-equation set-up, making it simple to implement and interpret.
• Different variables can be assigned different lag-lengths as they enter the model.

We need a road map to help us. Here are the basic steps that we're going to follow (with details to be added below):

1. Make sure than none of the variables are I(2), as such data will invalidate the methodology.
2. Formulate an "unrestricted" error-correction model (ECM). This will be a particular type of ARDL model.
3. Determine the appropriate lag structure for the model in step 2.
4. Make sure that the errors of this model are serially independent.
5. Make sure that the model is "dynamically stable".
6. Perform a "Bounds Test" to see if there is evidence of a long-run relationship between the variables.
7. If the outcome at step 6 is positive, estimate a long-run "levels model", as well as a separate "restricted" ECM.
8. Use the results of the models estimated in step 7 to measure short-run dynamic effects, and the long-run equilibrating relationship between the variables.

We can see from the form of the generic ARDL model given in equation (1) above, that such models are characterised by having lags of the dependent variable, as well as lags (and perhaps the current value) of other variables, as the regressors. Let's suppose that there are three variables that we're interested in modelling: a dependent variable, y, and two other explanatory variables, x₁ and x₂. More generally, there will be (k + 1) variables - a dependent variable, and k other variables.

Before we start, let's recall what a conventional ECM for cointegrated data looks like. It would be of the form:

          Δy_t = β₀ + Σ β_iΔy_t-i + Σγ_jΔx_1t-j + Σδ_kΔx_2t-k + φz_t-1 + e_t    ;        (2)

Here, z, the "error-correction term", is the OLS residuals series from the long-run "cointegrating regression",

           y_t = α₀ + α₁x_1t + α₂x_2t + v_t       ;       (3)

The ranges of summation in (2) are from 1 to p, 0 to q₁, and 0 to q₂ respectively.

Now, back to our own analysis-

Step 1:
We can use the ADF and KPSS tests to check that none of the series we're working with are I(2).

Step 2:

Formulate the following model:

    Δy_t = β₀ + Σ β_iΔy_t-i + Σγ_jΔx_1t-j + Σδ_kΔx_2t-k + θ₀y_t-1 + θ₁x_1t-1 + θ₂ x_2t-1 + e_t   ;    (4)

Notice that this is almost like a traditional ECM. The difference is that we've now replaced the error-correction term, z_t-1 with the terms y_t-1, x_1t-1, and x_2t-1. From (3), we can see that the lagged residuals series would be z_t-1 = (y_t-1 - a₀ - a₁x_1t-1 - a₂x_2t-1), where the a's are the OLS estimates of the α's. So, what we're doing in equation (4) is including the same lagged levels as we do in a regular ECM, but we're not restricting their coefficients.

This is why we might call equation (4) an "unrestricted ECM", or an "unconstrained ECM". Pesaran et al. (2001) call this a "conditional ECM".

Step 3:

The ranges of summation in the various terms in (4) are from 1 to p, 0 to q₁, and 0 to q₂ respectively.We need to select the appropriate values for the maximum lags, p, q₁, and q₂. Also, note that the "zero lags" on Δx₁ and Δx₂ may not necessarily be needed. Usually, these maximum lags are determined by using one or more of the "information criteria" - AIC, SC (BIC), HQ, etc. These criteria are based on a high log-likelihood value, with a "penalty" for including more lags to achieve this. The form of the penalty varies from one criterion to another. Each criterion starts with -2log(L), and then penalizes, so the smaller the value of an information criterion the better the result.

I generally use the Schwarz (Bayes) criterion (SC), as it's a consistent model-selector. Some care has to be taken not to "over-select" the maximum lags, and I usually also pay some attention to the (apparent) significance of the coefficients in the model.

Step 4:
A key assumption in the ARDL / Bounds Testing methodology of Pesaran et al. (2001) is that the errors of equation (4) must be serially independent. As those authors note (p.308), this requirement may also be influential in our final choice of the maximum lags for the variables in the model.

Once an apparently suitable version of (4) has been estimated, we should use the LM test to test the null hypothesis that the errors are serially independent, against the alternative hypothesis that the errors are (either) AR(m) or MA(m), for m = 1, 2, 3,...., etc.

Step 5:
We have a model with an autoregressive structure, so we have to be sure that the model is "dynamically stable". For full details of what this means, see my recent post, When is an Autoregressive Model Dynamically Stable? What we need to do is to check that all of the inverse roots of the characteristic equation associated with our model lie strictly inside the unit circle. That recent post of mine showed how to trick EViews into giving us the information we want in order to check that this condition is satisfied. I won't repeat that here.

Step 6:
Now we're ready to perform the "Bounds Testing"!

Here's equation (4), again:

      Δy_t = β₀ + Σ β_iΔy_t-i + Σγ_jΔx_1t-j + Σδ_kΔx_2t-k + θ₀y_t-1 + θ₁x_1t-1 + θ₂ x_2t-1 + e_t   ;    (4)

All that we're going to do is preform an "F-test" of the hypothesis, H₀:  θ₀ = θ₁ = θ₂ = 0 ; against the alternative that H₀  is not true. Simple enough - but why are we doing this?

As in conventional cointegration testing, we're testing for the absence of a long-run equilibrium relationship between the variables. This absence coincides with zero coefficients for y_t-1, x_1t-1 and x_2t-1 in equation (4). A rejection of H₀ implies that we have a long-run relationship.

There is a practical difficulty that has to be addressed when we conduct the F-test. The distribution of the test statistic is totally non-standard (and also depends on a "nuisance parameter", the cointegrating rank of the system) even in the asymptotic case where we have an infinitely large sample size.  (This is somewhat akin to the situation with the Wald test when we test for Granger non-causality in the presence of non-stationary data. In that case, the problem is resolved by using the Toda-Yamamoto (1995) procedure, to ensure that the Wald test statistic is asymptotically chi-square, as discussed here.)

Exact critical values for the F-test aren't available for an arbitrary mix of I(0) and I(1) variables. However, Pesaran et al. (2001) supply bounds on the critical values for the asymptotic distribution of the F-statistic. For various situations (e.g., different numbers of variables, (k + 1)), they give lower and upper bounds on the critical values. In each case, the lower bound is based on the assumption that all of the variables are I(0), and the upper bound is based on the assumption that all of the variables are I(1). In fact, the truth may be somewhere in between these two polar extremes.

If the computed F-statistic falls below the lower bound we would conclude that the variables are I(0), so no cointegration is possible, by definition. If the F-statistic exceeds the upper bound, we conclude that we have cointegration. Finally, if the F-statistic falls between the bounds, the test is inconclusive.

Does this remind you of the old Durbin-Watson test for serial independence? It should!

As a cross-check, we should also perform a "Bounds t-test" of H₀ : θ₀ = 0, against H₁ :  θ₀ < 0. If the t-statistic for y_t-1 in equation (4) is greater than the "I(1) bound" tabulated by Pesaran et al. (2001; pp.303-304), this would support the conclusion that there is a long-run relationship between the variables. If the t-statistic is less than the "I(0) bound", we'd conclude that the data are all stationary.

Step 7:
Assuming that the bounds test leads to the conclusion of cointegration, we can meaningfully estimate the long-run equilibrium relationship between the variables:

            y_t = α₀ + α₁x_1t + α₂x_2t + v_t       ;      (5)

as well as the usual ECM:

          Δy_t = β₀ + Σ β_iΔy_t-i + Σγ_jΔx_1t-j + Σδ_kΔx_2t-k + φz_t-1 + e_t    ;    (6)

where z_t-1 = (y_t-1 -a₀ - a₁x_1t-1 - a₂x_2t-1), and the a's are the OLS estimates of the α's in (5).

Step 8:
We can "extract" long-run effects from the unrestricted ECM. Looking back at equation (4), and noting that at a long-run equilibrium,  Δy_t = 0, Δx_1t = Δx_2t = 0, we see that the long-run coefficients for x₁ and x₂ are -(θ₁/ θ₀) and -(θ₂/ θ₀) respectively.

An Example:
Now we're ready to look at a very simple empirical example. I'm going to use the data for U.S. and European natural gas prices that I made available as a second example in my post, Testing for Granger Causality. I didn't go through the details of testing for Granger causality with that set of data, but I mentioned near the end of the post, and the EViews file (which included a "read_me" object with comments about the results) is there on the code page for this blog (dated 29 April, 2011).

If you look back at that earlier file, you'll find that I used the Toda-Yamamoto (1995) testing procedure to determine that there is Granger causality running from the U.S. series to the European series, but not vice versa.

A new EViews file that uses the same data for our ARDL modelling is available on the code page, under the date for the current post. The data for the two time-series we'll be using are also available on the data page for this blog. The data are monthly, from 1995(01) to 2011(03). In terms of the notation that was introduced earlier, we have (k + 1) = 2 variables, so k = 1 when it comes to the bounds testing.

Here's a plot of the data we'll be using (remember that you can enlarge most of these inserts by clicking on them):

To complete Step 1, we need to check that neither of our time-series are I(2). Applying the ADF test to the levels of EUR and US, the p-values are 0.53 and 0.10 respectively. Applying the test to the first-differences of the series, the p-values are both 0.00. (The lag-lengths for the ADF regressions were chosen using the Schwarz criterion, SC.) Clearly, neither series is I(2).

Applying the KPSS test we  reject the null of stationarity, even at the 1% significance level, for both EUR and US, but cannot reject the null of I(1) against I(2). The p-value of 10% for the ADF test of I(1) vs. I(0) for the EUR series may leave us wondering if that series is stationary, or not. You'll know that apparent "conflicts" between the outcomes of tests such as these are very common in practice.

This is a great illustration of how the ARDL / Bounds Testing methodology can help us. In order for standard cointegration testing (such as that of Engle and Granger, or Johansen) to make any sense, we must be really sure that all of the series are integrated of the same order. In this instance, you might not be feeling totally sure that this is the case.

Step 2 is straightforward. Given that the Granger causality testing associated with my earlier post suggested that there is causality from US to EUR (but not vice versa), ΔEUR is going to be the dependent variable in my unrestricted ECM:

          ΔEUR_t = β₀ + Σ β_iΔEUR_t-i + Σγ_jΔUS_t-j + θ₀EUR_t-1 + θ₁US_t-1 + e_t   ;     (5)

That's Step 2 out of the way!

To implement the information criteria for selecting the lag-lengths in an time-efficient way, I "tricked" EViews into providing lots of them at once by doing the following. I estimated a 1-equation VAR model for ΔEUR_t and I supplied the intercept, EUR_t-1, US_t-1, and a fixed number of lags of ΔUS_t as exogenous regressors. For example, when the fixed number of lags on ΔUS_t was zero, here's how I specified the VAR:

After estimating this model, I then chose VIEW, LAG STRUCTURE, LAG LENGTH CRITERIA:

I then repeated this by adding  ΔUS_t-1 to the list of exogenous variables, and got the following results:

I proceeded in this manner with additional lags of ΔUS_t in the "exogenous" list. I also considered cases such as:

which resulted in the following information criteria values:

Looking at the SC values in these three tables of results, we see that a maximum lag of 4 is suggested for ΔEUR_t. (The AIC values suggest that 8 lags of ΔEUR_t may be appropriate, but some experimentation with this was not fruitful.)

There is virtually no difference between the SC values for the case where the model includes just US_t as a regressor (0.8714), and the case where just ΔUS_t-1 is included (0.8718). To get some dynamics into the model, I'm going to go with the latter case.

With Step 3 completed, and with this lag specification in mind, let's now look at the estimated unrestricted ECM:

Step 4 involves checking that the errors of this model are serially independent. Selecting VIEW, RESIDUAL DIAGNOSTICS, SERIAL CORRELATION LM TEST, I get the following results:

m           LM          p-value

1           0.079        0.779

2           2.878        0.237

3           5.380        0.146

4         11.753        0.019

O.K., we have a problem with serial correlation! To deal with it, I experimented with one or two additional lags of the dependent variable as regressors, and ended up with the following specification for the unrestricted ECM:

The serial independence results now look much more satisfactory:

m           LM          p-value

1           0.013        0.911

2           3.337        0.189

3           5.183        0.159

4           7.989        0.092

5           8.473        0.132

6         11.023        0.088

7         12.270        0.092

8         12.334        0.137

Next, Step 5 involves checking the dynamic stability of this ARDL model. Here are the inverse roots of the associated characteristic equation:

All seems to be well - these roots are all inside the unit circle.

Before proceeding to the Bounds Testing, let's take a look at the "fit" of our unrestricted ECM. The "Actual / Fitted / Residuals" plot looks like this:

When we "unscramble" these results, and look at the fit of the model in terms of explaining the level of EUR itself, rather than ΔEUR, things look pretty good: 

We're now ready for Step 6 - the Bounds Test itself. We want to test if the coefficients of both EUR(-1) and US(-1) are zero in our estimated model (repeated below):

 

The associated F-test is obtained as follows:

With the result:

The value of our F-statistic is 5.827, and we have (k + 1) = 2 variables (EUR and US) in our model. So, when we go to the Bounds Test tables of critical values, we have k = 1.

Table CI (iii) on p.300 of Pesaran et al. (2001) is the relevant table for us to use here. We haven't constrained the intercept of our model, and there is no linear trend term included in the ECM.  The lower and upper bounds for the F-test statistic at the 10%, 5%, and 1% significance levels are [4.04 , 4.78], [4.94 , 5.73], and [6.84 , 7.84] respectively. 

As the value of our F-statistic exceeds the upper bound at the 5% significance level, we can conclude that there is evidence of a long-run relationship between the two time-series (at this level of significance or greater). 

In addition, the t-statistic on EUR(-1) is -2.926. When we look at Table CII (iii) on p.303 of Pesaran et al. (2001), we find that the I(0) and I(1) bounds for the t-statistic at the 10%, 5%, and 1% significance levels are [-2.57 , -2.91], [-2.86 , -3.22], and [-3.43 , -3.82] respectively. At least at the 10% significance level, this result reinforces our conclusion that there is a long-run relationship between EUR and US.

So, here we are at Step 7 and Step 8.

Recalling our preferred unrestricted ECM:

we see that the long-run multiplier between US and EUR is -(0.047134 / (-0.030804)) = 1.53. In the long run, an increase of 1 unit in US will lead to an increase of 1.53 units in EUR.

If we estimate the levels model,

                   EUR_t = α₀ + α₁US_t + v_t       ,

by OLS, and construct the residuals series, {z_t}, we can fit a regular (restricted) ECM:

Notice that the coefficient of the error-correction term, z_t-1, is negative and very significant. This is what we'd expect if there is cointegration between EUR and US. The magnitude of this coefficient implies that nearly 3% of any disequilibrium between EUR and US is corrected within one period (one month).

This final ECM is dynamically stable:

As none of the roots lie on the X (real) axis, it's clear that we have three complex conjugate pairs of roots. Accordingly, the short-run dynamics associated with the model are quite complicated. This can be seen if we consider the impulse response function associated with a "shock" of one (sample) standard deviation:

Finally, the within-sample fit (in terms of the levels of EUR) is exceptionally good:

In fact, the simple correlations between EUR and the "fitted" EUR series from the unrestricted and regular ECM's are each 0.994, and the correlation between the two fitted series is 0.9999.

So, there we have it - bounds testing with an ARDL model.

[Note: For an important update of this post, relating to EViews 9, see my 2015 post, here.]

References

Pesaran, M. H. and Y. Shin, 1999. An autoregressive distributed lag modelling approach to cointegration analysis. Chapter 11 in S. Strom (ed.), Econometrics and Economic Theory in the 20th Century: The Ragnar Frisch Centennial Symposium. Cambridge University Press, Cambridge. (Discussion Paper version.)

Pesaran, M. H., Shin, Y. and Smith, R. J., 2001. Bounds testing approaches to the analysis of level relationships. Journal of Applied Econometrics, 16, 289–326.

Pesaran, M. H. and R. P. Smith, 1998. Structural analysis of cointegrating VARs. Journal of Economic Surveys, 12, 471-505.

Toda, H. Y and T. Yamamoto (1995). Statistical inferences in vector autoregressions with possibly integrated processes. Journal of Econometrics, 66, 225-250.

© 2013, David E. Giles